Phi and An Elegant Equation

Let me clarify. I was on vacation 20 years ago when I discovered an interesting and, in my opinion, elegant mathematical equation involving Phi.

Life intervened and 20 years later, while perusing one of my personal journals in which I keep math and physics thoughts, I stumbled upon the equation.

So I sat down and wrote a proof.

As far as I know it is original. I have researched and not found this published anywhere.

Note: This is not a story about Phi or the Golden Ratio. More on Phi and its relation to the Golden Ratio can be found here.

It is an equation I discovered, and the proof I subsequently wrote. The elegance of this equation has stuck in my mind for 20 years and I’d like to share with you.

Equation and Proof

Computer Program

If you want to play with it, I have included a small JavaScript computer program. I wrote this in VSCode.


As stated at GoldenNumber.Net,

Phi ( Φ = 1.618033988749895… ), most often pronounced fi like “fly,” is simply an irrational number like pi ( p = 3.14159265358979… ), but one with many unusual mathematical properties. Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation

Phi has many interesting properties and this appears to be another. I have tried something similar Things with PI, e and other irrational and transcendental numbers, but have not been successful in deriving any special equations. I find this an elegant little equation and will be experimenting with it further.

Thanks for reading!

Self-Employed Software Developer, Trainer, Consultant. Keeping up to date. I’ve noticed in over 28 years of programming, one’s current skills have a shelf life.

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